High-Frequency Modeling and Analyses for Buck and Multiphase Buck Converters

140 Pages·2005·2.12 MB·English

by  Yang Qiu

High-Frequency Modeling and Analyses for
Buck and Multiphase Buck Converters

Yang Qiu

Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of

Doctor of Philosophy
in
Electrical Engineering

APPROVED
Fred C. Lee, Chairman
Daan van Wyk
Ming Xu
Yilu Liu
Guo-Quan Lu

November 30th, 2005
Blacksburg, Virginia

Keywords: high frequency, sideband effect, multi-frequency model,
buck converter, multiphase

© 2005, Yang Qiu
High-Frequency Modeling and Analyses for
Buck and Multiphase Buck Converters
Yang Qiu
(Abstract)
Future microprocessor poses many challenges to its dedicated power supplies, the
voltage regulators (VRs), such as the low voltage, high current, fast load transient, etc. For
the VR designs using multiphase buck converters, one of the results from these stringent
challenges is a large amount of output capacitors, which is undesired from both a cost and
a motherboard real estate perspective. In order to save the output capacitors, the control-
loop bandwidth must be increased. However, the bandwidth is limited in the practical
design. The influence from the switching frequency on the control-loop bandwidth has not
been identified, and the influence from multiphase is not clear, either. Since the widely-
used average model eliminates the inherent switching functions, it is not able to predict the
converter’s high-frequency performance. In this dissertation, the primary objectives are to
develop the methodology of high-frequency modeling for the buck and multiphase buck
converters, and to analyze their high-frequency characteristics.
First, the nonlinearity of the pulse-width modulator (PWM) scheme is identified.
Because of the sampling characteristic, the sideband components are generated at the
output of the PWM comparator. Using the assumption that the sideband components are
well attenuated by the low-pass filters in the converter, the conventional average model
only includes the perturbation-frequency components. When studying the high-frequency
performance, the sideband frequency is not sufficiently high as compared with the
perturbation one; therefore, the assumption for the average model is not good any more.
Under this condition, the converter response cannot be reflected by the average model.
Furthermore, with a closed loop, the generated sideband components at the output voltage
appear at the input of the PWM comparator, and then generate the perturbation-frequency
components at the output. This causes the sideband effect to happen. The perturbation-
frequency components and the sideband components are then coupled through the
comparator. To be able to predict the converter’s high-frequency performance, it is
necessary to have a model that reflects the sampling characteristic of the PWM
comparator. As the basis of further research, the existing high-frequency modeling
approaches are reviewed. Among them, the harmonic balance approach predicts the high-
frequency performance but it is too complicated to utilize. However, it is promising when
simplified in the applications with buck and multiphase buck converters. Once the
nonlinearity of the PWM comparator is identified, a simple model can be obtained because
the rest of the converter system is a linear function.
With the Fourier analysis, the relationship between the perturbation-frequency
components and the sideband components are derived for the trailing-edge PWM
comparator. The concept of multi-frequency modeling is developed based on a single-
phase voltage-mode-controlled buck converter. The system stability and transient
performance depend on the loop gain that is affected by the sideband component. Based on
the multi-frequency model, it is mathematically indicated that the result from the sideband
effect is the reduction of magnitude and phase characteristics of the loop gain. With a
higher bandwidth, there are more magnitude and phase reductions, which, therefore, cause
the sideband effect to pose limitations when pushing the bandwidth.
The proposed model is then applied to the multiphase buck converter. For voltage-
mode control, the multiphase technique has the potential to cancel the sideband effect
around the switching frequency. Therefore, theoretically the control-loop bandwidth can be
pushed higher than the single-phase design. However, in practical designs, there is still
magnitude and phase reductions around the switching frequency in the measured loop gain.
Using the multi-frequency model, it is clearly pointed out that the sideband effect cannot
be fully cancelled with unsymmetrical phases, which results in additional reduction of the
phase margin, especially for the high-bandwidth design. Therefore, one should be
extremely careful to push the bandwidth when depending on the interleaving to cancel the
sideband effect.
The multiphase buck converter with peak-current control is also investigated.
Because of the current loop in each individual phase, there is the sideband effect that
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cannot be canceled with the interleaving technique. For higher bandwidths and better
transient performances, two schemes are presented to reduce the influence from the current
loop: the external ramps are inserted in the modulators, and the inductor currents are
coupled, either through feedback control or by the coupled-inductor structure. A bandwidth
around one-third of the switching frequency is achieved with the coupled-inductor buck
converter, which makes it a promising circuit for the VR applications.
As a conclusion, the feedback loop results in the sideband effect, which limits the
bandwidth and is not included in the average model. With the proposed multi-frequency
model, the high-frequency performance for the buck and multiphase buck converters can
be accurately predicted.
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TO MY PARENTS

FEIZHOU QIU AND CUIZHEN LIU

AND TO MY WIFE

JUANJUAN SUN
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Acknowledgments
I would like to express my sincere appreciation to my advisor, Dr. Fred C. Lee, for
his continued guidance, encouragement and support. It is an honor to be one of his students
here at the Center for Power Electronics Systems (CPES), one of the best research centers
in power electronics. In the past years, I am always amused by his great intuition, broad
knowledge and accurate judgment. The most precious things I learned from him are the
ability of independent research and the attitude toward research, which can be applied to
every aspects of life and will benefit me for the rest of my life.
I would also like to thank Dr. Ming Xu for his enthusiastic help during my research
at CPES. His selfless friendship and leadership helped to make my time at CPES enjoyable
and rewarding. From him, I learned so much not only in the knowledge of power
electronics but also in the research methodologies. His valuable suggestions helped to
encourage my pursuing this degree.
I am grateful to the other members of my advisory committee, Dr. Daan van Wyk,
Dr. Yilu Liu, Dr. Guo-Quan Lu, and Dr. Dan Y. Chen for their support, comments,
suggestions and encouragement.
I am especially indebted to my colleagues in the VRM group and the ARL group. It
has been a great pleasure to work with the talented, creative, helpful and dedicated
colleagues. I would like to thank all the members of my teams: Dr. Peng Xu, Dr. Pit-Leong
Wong, Dr. Kaiwei Yao, Dr. Wei Dong, Dr. Francisco Canales, Dr. Bo Yang, Dr. Jia Wei,
Mr. Mao Ye, Dr. Jinghai Zhou, Dr. Yuancheng Ren, Mr. Bing Lu, Mr. Yu Meng, Mr.
Ching-Shan Leu, Mr. Doug Sterk, Mr. Kisun Lee, Mr. Julu Sun, Dr. Shuo Wang, Dr. Xu
Yang, Mr. Yonghan Kang, Mr. Chuanyun Wang, Mr. Dianbo Fu, Mr. Arthur Ball, Mr.
Andrew Schmit, Mr. David Reusch, Mr. Yan Dong, Mr. Jian Li, Mr. Bin Huang, Mr. Ya
Liu, Mr. Yucheng Ying, and Mr. Yi Sun. It was a real honor working with you guys.
I would like to thank my fellow students and visiting scholars for their help and
guidance: Dr. Peter Barbosa, Mr. Dengming Peng, Dr. Jinjun Liu, Dr. Jae-Young Choi, Dr.
Qun Zhao, Dr. Zhou Chen, Dr. Jinghong Guo, Dr. Linyin Zhao, Dr. Rengang Chen, Dr.
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Zhenxue Xu, Dr. Bin Zhang, Dr. Xigen Zhou, Ms. Qian Liu, Mr. Xiangfei Ma, Mr. Wei
Shen, Dr. Haifei Deng, Ms. Yan Jiang, Ms. Huiyu Zhu, Mr. Pengju Kong. Mr. Jian Yin,
Mr. Wenduo Liu, Dr. Zhiye Zhang, Ms. Ning Zhu, Ms. Jing Xu, Ms. Yan Liang, Ms.
Michele Lim, Mr. Chucheng Xiao, Mr. Hongfang Wang, Mr. Honggang Sheng, and Mr.
Rixin Lai.
I would also like to thank the wonderful members of the CPES staff who were
always willing to help me out, Ms. Teresa Shaw, Ms. Linda Gallagher, Ms. Teresa Rose,
Ms. Ann Craig, Ms. Marianne Hawthorne, Ms. Elizabeth Tranter, Ms. Michelle
Czamanske, Ms. Linda Long, Mr. Steve Chen, Mr. Robert Martin, Mr. Jamie Evans, Mr.
Dan Huff, Mr. Callaway Cass, Mr. Gary Kerr, and Mr. David Fuller.
My heartfelt appreciation goes toward my parents, Feizhou Qiu and Cuizhen Liu,
who have always provided support and encouragement throughout my further education.
Finally, with deepest love, I would like to thank my wife, Juanjuan Sun, who has
always been there with her love, support, understanding and encouragement for all of my
endeavors.
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This work was supported by the VRM consortium (Artesyn, Delta Electronics, Hipro
Electronics, Infineon, Intel, International Rectifier, Intersil, Linear Technology, National
Semiconductor, Renesas, and Texas Instruments), and the Engineering Research Center
Shared Facilities supported by the National Science Foundation under NSF Award Number
EEC-9731677. Any opinions, findings and conclusions or recommendations expressed in
this material are those of the author and do not necessarily reflect those of the National
Science Foundation.
This work was conducted with the use of SIMPLIS software, donated in kind by
Transim Technology of the CPES Industrial Consortium.
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Table of Contents
Chapter 1. Introduction.......................................................................................................1
1.1 Background: Voltage Regulators..............................................................................1
1.2 Challenges to VR High-Frequency Modeling..........................................................6
1.3 Dissertation Outlines..............................................................................................12
Chapter 2. Characteristics of PWM Converters.............................................................14
2.1 Introduction.............................................................................................................14
2.2 Characteristics of the Pulse-Width Modulator........................................................15
2.3 Sideband Effect of PWM Converters with Feedback Loop...................................23
2.4 Small-Signal Transfer Function Measurements and Simulations..........................32
2.5 Previous Modeling Approaches..............................................................................34
2.6 Summary.................................................................................................................37
Chapter 3. Multi-Frequency Modeling for Buck Converters........................................39
3.1 Modeling of the PWM Comparator........................................................................39
3.2 The Multi-Frequency Model of Buck Converters..................................................45
3.3 Summary.................................................................................................................52
Chapter 4. Analyses for Multiphase Buck Converters...................................................54
4.1 Introduction.............................................................................................................54
4.2 The Multi-Frequency Model of Multiphase Buck Converters...............................55
4.3 Study for the Multiphase Buck Converter with Unsymmetrical Phases................66
4.4 Summary.................................................................................................................71
Chapter 5. High-Bandwidth Designs of Multiphase Buck VRs with Current-Mode
Control................................................................................................................................73
5.1 Introduction.............................................................................................................73
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5.2 Bandwidth Improvement with External Ramps .....................................................78
5.3 Bandwidth Improvement with Inductor Current Coupling.....................................81
5.4 Summary.................................................................................................................96
Chapter 6. Conclusions......................................................................................................97
6.1 Summary.................................................................................................................97
6.2 Future Works..........................................................................................................99
Appendix A. Analyses with Different PWM Schemes..................................................100
Appendix B. Analyses with Input-Voltage Variations..................................................109
References.........................................................................................................................117
Vita....................................................................................................................................122

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